Coherent States in Quantum Mechanicsscipp.ucsc.edu/~haber/ph251/Rockwood_Presentation.pdf · Gavin Rockwood UCSC Coherent States in Quantum Mechanics June 14, 2019 16 / 27. Coherent

Coherent States of the Quantum Harmonic Oscillator General Coherent States Applications References

Coherent States in Quantum Mechanics

Gavin Rockwood

June 14, 2019

Gavin Rockwood UCSC Coherent States in Quantum Mechanics June 14, 2019 1 / 27


Overview

1 Coherent States of the Quantum Harmonic OscillatorIntroductionMinimal Uncertainty StatesThe Displacement Operator

2 General Coherent StatesDefinitionHeisenberg-Weyl GroupCoherent Spin States

3 ApplicationsCoherent States in OpticsCoherent States in Superfluids



Coherent States of the Quantum Harmonic Oscillator



Introduction

The Quantum Harmonic Oscillator

For our discussion, we will deal with the harmonic oscillator in 1 dimension!

The Hamiltonian

Ĥ = −~2P̂2

2m+

12

mωX̂ 2 (1)

â :=1

√2~mω

(mωQ̂ + i P̂

)â† :=

1√

2~mω

(mωQ̂ − i P̂

) (2)=⇒ Ĥ = ~ω

(â†â +

12

)(3)



Minimal Uncertainty States


For the QHO, lets calculate ∆X∆P.

Show ∆X∆P =Starting with:

X̂ =

√~

2mω

(â + â†

)P̂ = −i

√~mω

2

(â− â†

) (4)With this we have:

∆x∆p =√〈n| X̂ 2 |n〉 − 〈n| X̂ |n〉2

√〈n| P̂2 |n〉 − 〈n| P̂ |n〉2 (5)

=~2

(2n + 1) (6)

We see that n = 0 saturates the Heisenberg Uncertainty relation: ∆x∆p ≥ ~2 .




Another Minimal Uncertainty State

Lets define the following state:

|z〉 = e−12 |z|

2∞∑

n=0

zn√

n!|n〉 ; z ∈ C (7)

This state is exciting because â |z〉 = z |z〉. Also 〈z|z〉 = 1. Its good to note that |z〉can also be written as:

|z〉 = e−12 |z|

2+zâ |0〉 (8)

Note!It is worth noting that there is a yet unmentioned way to write |z〉 that will be discussedshortly




Another Minimal Uncertainty State Continued

As mentioned, |z〉 is a minimal uncertainty state. To see this:

Proof that |z〉 is a minimal uncertainty state.

Starting with:

〈z| X̂ |z〉 =√

~2mω

(〈z| â |z〉+ 〈z| â† |z〉

)=

√~mω

2(z + z∗)

〈z| P̂ |z〉 = −i√

~mω2

(z − z∗)

〈z| X̂ 2 |z〉 =√

~2mω

((z + z∗)2 + 1

)〈2| P̂2 |z〉 = sqrt

~mω2

((z − z∗)2 − 1

)(9)

With this:∆x∆p =

~2

(10)



The Displacement Operator

The Translation Operator

Let us define the following operator, called the Heisenberg-Weyl Translation operator.

T̂ (z) = ei~ (pQ̂−qP̂),

1√

2~mω(mωq + ip) ∈ C (11)

This operator can also be written in terms of creation and annihilation operators:

T̂ (z) = ezâ†−z∗â (12)

This last version is the one most commonly seen in discussions about coherent states.

We will go back and explore the Heisenberg-Weyl group in greater depth when we talkabout general coherent states




The Action of T̂ (z) on Q̂ and P̂

We are after T̂ (z)Q̂T̂−1(z) and T̂ (z)P̂T̂−1(z).

Start with T̂ (z)âT̂−1(z).

T̂ (z)âT̂−1(z) = â−[â, zâ† − z∗â

]= â− z

By expanding â, z in terms of Q̂, P̂, q, p we get:

T̂ (z)Q̂T̂−1(z) = Q̂ − q

T̂ (z)P̂T̂−1(z) = P̂ − p(13)




|z〉 from D̂(z)

Earlier I mentioned that there was another way to write |z〉 and that is as:

|z〉 = D̂(z) |0〉 (14)

Some Quick Observations

If we let φz be the position wavefunction of |z〉 then:

T̂ (0) |0〉 = |0〉 =⇒ T̂ (0)φ0 = φ0 where φ0 is the groundstate wavefunction of theharmonic oscillator. In other words, the groundstate is a coherent state.

T̂ (z)φ0(x) = e− i2~ qpe

i~ xpφ0(x − q)

T̂ (z)F [φ0](k) = e−i

2~ qpei~ qkF [φ0](k − p)

(15)

Where F [φ0](k) is the Fourier transform of φ0. Here, we see that |z〉 is nowlocalized at Q̂ = q and P̂ = p in phase space.




Time Evolution!

The time evolution of T̂ (z) is given by:

T̂ (zt ) = e−iHHOt/~T̂ (z0)eiHHOt/~ (16)

With this, the time evolution of a coherent state is given by:

e−iHHOt/~ |z0〉 = |z(t)〉 , z(t) = eiωt z0 (17)

Time Evolution of 〈Q〉 and 〈P〉

〈z(t)| Q̂ |z(t)〉 = p0 sin(ωt)− q0 cos(ωt)

〈z(t)| P̂ |z(t)〉 = −q0 sin(ωt)− p0 cos(ωt)(18)



General Coherent States



Definition

Definition of General Coherent State

Let G be an arbitrary Lie group and T be an irreducible representation acting on a Hilbertspace H. Pick a vector |ψ0〉 ∈ H, then the coherent state system {T , |ψ0〉} is the set{|ψ〉 ∈ H s.t. |ψ〉 = T (g) |ψ0〉 , g ∈ G}.

Now, we will want to look at a subgroup H ={

g ∈ G s.t. T (g) |ψ0〉 = eiα(g) |ψ0〉}

. H isthe isotropy group of |ψ0〉.

DefinitionThen a general coherent state |ψg〉 = g |ψ0〉 is determined by a point x = x(g) in thecoset space G/H such that |ψg〉 = eiα(x) |x〉.



Heisenberg-Weyl Group

Looking at the Heisenberg - Weyl Group

They algebra of the Heisenberg-Weyl group in n dimensions (denoted as Wn is generatedby I, Q1, ...Qn and P1, ...Pn.[

Qi ,Qj]

=[Pi ,Pj

]= [I,Qi ] = [I,Pi ] = 0[

Qi ,Pj]

= δij~I(19)

The Lie Algebra hn is generated by I → i Î, Q → iQ̂ and P → i P̂. Thus, any element ofhn can be written as:

Ŵ =it2~

I+i~

(p · Q̂ − q · P̂

)=

it2~

+i~

L̂(z), L̂(z) :=(

p · Q̂ − q · P̂), z = (q, p) ∈ R2n

(20)So, eŴ = e

it2~ e

i~ L̂(z) = e

it2~ T̂ (z)




Some Properties of T̂

The product of T̂ (z) and T̂ (z′).

T̂ (z)T̂ (z′) = e−i

2~ (q·p′−p·q′)T̂ (z + z′) ∈ Wn (21)

The identity operator is eŴ (t=0,z=0) and for a given state |ψ〉, the isotropy group is givenby Hn =

{eŴ (t,0)

}∼= U(1).

The set of coherent operators is Wn/Hn ={T̂ (z), z = (q, p) ∈ R2n

}




So Back to the Harmonic Oscillator

If we go back to the harmonic oscillator, we had a ground state |0〉, The group we areconsidering is W1, the isotropy group is still

{eŴ (t=0,z=0

}∼= U(1). Thus coherent

states of the Heisenberg-Weyl group are generated by T (z), z = (q, p) ∈ R2.




A Few More Notes on the Harmonic Oscillator

The set{

T̂ (z) |0〉}

is complete but not necessarily orthogonal:

〈z′|z

〉= 〈0| T̂−1(z′)T̂ (z) |0〉 (22)= 〈0|T (−z′)T (z) |0〉 (23)

= ei

2~ (q′p−p′q) 〈0| T̂ (z′ + z) |0〉 (24)

= exp

[i

2~(q′p − p′q

)−

(q′ − q)2 + (p′ − p)2

4~

](25)

6= 0 (26)

This last part comes from the fact 〈0| T̂ (z) |0〉 = exp(− q

2+p2

4~

). This implies{

T̂ (z) |0〉}

is in fact overcomplete.



Coherent Spin States


Let |j,m〉 be a state of spin j and z projection m. Then J3 |j,m〉 = m |j,m〉 andJ2 |j,m〉 = j(j + 1) |j,m〉. The uncertainty relation ∆J1∆J2 ≥ 12 〈J3〉.

∆J1∆J2 =12

(j(j + 1)−m2

)(27)

States of minimal uncertainty are given by m = ±j .




Sooo, whats G and H?

Our Lie group for this system is G = SU(2). With this, we can go back to our definitionof a general coherent state. The isotropy group is U(1) and thus the phase space isSU(2)/U(1) which is topologically equivalent to S2.

Now! Lets construct them!




Constructing Spin Coherent States Defining Some Ladder Operators

Lets start by defining:

Ladder Operators!

Ĵ± = Ĵ1 ± i Ĵ2 (28)

whereĴ± |j,m〉 =

√(j ∓m)(j ±m + 1) |j,m ± 1〉 (29)

and [Ĵ3, Ĵ±

]= ±Ĵ±[

Ĵ+, Ĵ−]

= 2Ĵ3(30)




Ok, Now Start to Construct Spin Coherent States

We begin by defining an operator:

D̂(ξ) = eξĴ+−ξ∗ Ĵ− , ξ = − tan

(θ

2

)e−iφ (31)

Then, in a similar fashion as earlier, a coherent state is defined as:

|ξ〉 = D̂(ξ) |j,−j〉 =j∑

m=−j

[(2j)!

(j + m)(j −m)

] 12

(1 + |ξ|2)−jξj+µ |j,m〉 (32)



Applications



Coherent States in Optics

The Squeeze Operator

The squeeze operator is defined as:

S(ζ) := e12

(ζ∗a2−ζa†2

), ζ = reiθ (33)



Coherent States in Optics

Why are These Useful?

Optical Communication: Helps improve signal to noise ratio

Interferometry: LIGO!

Some weird quantum information stuff



Coherent States in Superfluids

Coherent States of Fields

Superfluids are discussed in terms of bosonic fields. So with that in mind, lets define:

|zk 〉 = e−12 |z|

2∞∑nk

znk√nk !|nk 〉 (34)

Where nk is the number of states in the k th mode of the field. We have the completenessrelation:

1π

∫dzk |zk 〉〈zk | = 1 (35)

The set of all states of the form∏

k |zk 〉 form a complete set of states.




Free Energy of |{z}〉

The free energy of |{z}〉 is given by:

F {z} = −kb ln[〈{z}| exp

(−

Ĥ(0) − µN̂kbT

)|{z}〉

](36)

Which to first order goes like:

F{0}{z} = −kb∑

k

e− 12 ~ω−µkbT |zk |2 (37)

This is a good approximation for a Bose-Einstein condensate. Adding a quadraticinteraction term to 36 will approximate the free energy of a superfluid because it allowsthe BEC to "flow".




[1] J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams,T. Adams, P. Addesso, R. X. Adhikari, and et al. Enhanced sensitivity of the LIGOgravitational wave detector by using squeezed states of light. Nature Photonics,7:613–619, August 2013.

[2] Monique Combescure and Didier Robert. Coherent States and Applications inMathematical Physics, volume 2012. 02 2012.

[3] K. Fujii. Introduction to Coherent States and Quantum Information Theory. eprintarXiv:quant-ph/0112090, December 2001.

[4] J. S. Langer. Coherent states in the theory of superfluidity. Phys. Rev., 167:183–190,Mar 1968.

[5] A. I. Lvovsky. Squeezed light. arXiv e-prints, January 2014.

[6] A. Perelomov. Generalized coherent states and their applications.

[7] Henning Vahlbruch, Moritz Mehmet, Karsten Danzmann, and Roman Schnabel.Detection of 15 db squeezed states of light and their application for the absolutecalibration of photoelectric quantum efficiency. Phys. Rev. Lett., 117:110801, Sep2016.

[8] D. F. Walls. Squeezed states of light. Nature, 306:141–146, 1983.


Coherent States of the Quantum Harmonic OscillatorIntroductionMinimal Uncertainty StatesThe Displacement Operator

General Coherent StatesDefinitionHeisenberg-Weyl GroupCoherent Spin States

ApplicationsCoherent States in OpticsCoherent States in Superfluids

References

Documents

Coherent States in Quantum Mechanicsscipp.ucsc.edu/~haber/ph251/Rockwood_Presentation.pdf · Gavin Rockwood UCSC Coherent States in Quantum Mechanics June 14, 2019 16 / 27. Coherent